Last year, I made the same bet all around the United States with every crowd of math teachers I met:

I’ll pick a number between 1 and 100. I’ll give you ten guesses to figure out my number. And every time you guess, I’ll tell you if my number is higher or lower.

I always wagered whatever cash I had in my pocket – generally between $2 to $20. The math teachers, meanwhile, owed me nothing if they lost. I had no trouble finding people to take the other side of that wager.

Then 83. I’m lower. Between 81 and 83. Four guesses left, but she only needs one. The crowd smells blood.

Then, with a trace of sympathy in her voice, 82. The crowd thinks it’s over.

But I’m higher.

Aaaand the chase is back on, y’all!.

Tentatively now: 82.5. I’m still higher. One by one, members of the crowd are wise to my scam.

Then 82.75. I’m lower. She has one guess left.

Then 82.7. I’m higher, at 82.72.

I asked her what I’d ask any crowd of sixth graders at this point:

If I offered you the same wager again, what follow-up questions would you have for me?

“What kind of number are you picking?” she said.

My point in all of this is that math teachers have names for their numbers, much in the same way that ornithologists have names for their birds. And much in the same way that ornithologists haven’t given me a reason to care about the difference between a Woodlark and a Skylark, math teachers often fail to motivate the difference between rational numbers and integers and whole numbers and imaginary numbers and supernatural numbers.

The difference is that ornithology isn’t a course that’s required for high school graduation and university enrollment and labor market participation. Kids aren’t forced to study ornithology for twelve years of their childhood.

So I’m inviting us to ask ourselves: “Why did we invent these categories of numbers?” And if we agree that it was to more effectively communicate about numbers, we need to put students in a place where their communication suffers without those categories. If we can’t, then we should confess those categories are vanity.

Before we give students the graphic organizers and Venn diagrams and foldables designed to help them learn those categories, let’s help them understand that they were invented for a reason. Not because we have to.

There are always ways to make kids memorize disconnected, purposeless stuff.

But because we should.

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Via email:

Did you ever lose?

I never once lost. I was never once asked to specify the kind of number I was picking.

Dan Meyer

Theresa

Anytime I need to settle an argument (say, who gets to go first), I do the whole pick a number between x and y bit. At the start of the year, i pretty much always choose 0 and 1 as my bounds, and students are confused at first, but always eventually get it. Then we get to have a quick fun discussion about comparing their choices to mine to figure out who gets to go first. (Is it odd that I frequently have to settle arguments about who gets to go first?)

Dan Meyer

Have you ever seen the neat little cartoon The Weird Number? It’s floating around on youtube, but it’s a terrible copy. I just can’t convince myself to fork out $65 for a 12 minut4e video, so I keep using youtube.

Anyway, it’s a neat little video originally made for second graders. Witty in a goofy way.

I show that every year to my algebra 2 and precalc classes, and sometimes trig. It’s totally appropriate for earlier classes, I just don’t teach anything.

The other thing I do is tell a story of (pick name of kid in class), who has sheep or goats. And we talk about his expanding empire of sheep, or goats, and how developments lead to different sorts of numbers. So addition and multiplication are closed for counting numbers (and everything beyond). Samir is perfectly happy counting his sheep. But then Alondra (another class member) realizes hey, I have NO sheep. Talk about the meaning of zero and how it wasn’t always obvious. So Alondra borrows 2 sheep from Samir to start her own crop. That works great for Alondra, who can count her sheep, but how can we represent Samir and Alondra’s sheep empire together? Then Samir dies with 857 sheep and 42 heirs who he wants to inherit equally. Hmm.

From there, we say goodbye to Samir and Alondra and turn to Euclid (who wasn’t first, but it links their minds to geometry). Although now that I think about it, I could probably continue with them. I usually do it as an area problem, not a isosceles triangle. So “Square root” is the root side of a square. So root side of an area 9 is 3, root side of area 4 is 2, but what’s the root side of a square with area 2?

Yep – this is it.

These are fully intended to be simplistic examples (and certainly not perfect history), but the point is that totally ordinary needs drove mathematicians to think about and develop new expanding number categories. Each one expanded our notion of reality, just like the Weird Number changed the town’s understanding of who they could be.

(I also ask the kids to identify which number category the movie is missing).

I usually do this at some point before I introduce i.

Joanne Ward

Love this post! By all means, we should change the ways we assess our students, too, especially those important tests that will significantly determine students’ future, for example: SAT, SAT Math I, II, or GRE.

I totally agree with you, but I also struggle almost at daily basis simply because I also want my students to do well on their standardized tests before my instructional skills have improved at a point that I can fully get rid of memorization.

What you said is my ultimate goal as a math teacher and something I need to sleep with probably the whole time as a math teacher. I really want to be a math teacher that teaches math.

Thank you for writing this article. I will keep this in my mind. I am a math teacher and this is what I need to do for my students.

William Ayres

Was math invented or discovered? Seemingly, the rational numbers were invented as a means to count things. But the irrational numbers also exist in many instances, most notably circles. How far does a wheel roll in one revolution for a given diameter? So maybe the irrational numbers were invented as a means to measure things?

Let’s compare Pythagorean tuning to well-tempered tuning in music. The rational versus the irrational. They each create effective harmony but one generates an audible tone outside the chord. Yet without the other we would not have music as we now understand it. Is music a way to discover the answer to the invented versus discovered? Or is it just more noise in the whole discussion? It seems that there is much that we don’t know.

Alistair Windsor

Unfortunately the names that we give to classes of numbers are inconsistent. The natural numbers are just a bust at this point – you have to ask whether 0 is in or not. The classes you list “rational numbers and integers and whole numbers and imaginary numbers and supernatural numbers” are generally OK though some author use imaginary numbers for complex numbers and pure imaginary number for what I would call imaginary. The point is that I find the focus on “academic vocabulary” to be a bit of a distraction.

Dan Meyer

Chester Draws

How are students meant to think about something if they haven’t got a word for the concept. Far from a distraction, focus on vocabulary in Maths is vital! I would have thought this one area that everyone, from old style traditional to most progressive, agreed on.

How does factorising work if students are unclear what a factor is? And how to explain we want whole number factors to someone who doesn’t know what a whole number is?

Dan Meyer

Joanne Ward

Dan Meyer

Just so there’s no confusion about my comment and post, I think academic vocabulary is useful and that we should teach it.

But students don’t need to know the word “linear” to think in depth about linear patterns. They don’t need to know the word “vertex” to reason about the apex of a basketball’s height.

I’m not saying never teach that vocabulary, rather I’m acknowledging that students have loads of informal ways of reckoning with those concepts that make the eventual instruction of that vocabulary more effective and interesting.

I’m saying, let’s cut it with the lengthy word lists that start off the chapter.

Dick Fuller

“How are students meant to think about something if they haven’t got a word for the concept?”

What is the word for the concept “to think about a concept requires a word for it”. I don’t know the word but I can think about the concept, and I hope we can talk about it. I confess I can’t see how mathematics, including the learning of it, can happen if the word for a concept must precede the concept. Do children need the name for a counting concept before they count. I don’t know a name for a concept for natural numbers, my “concept” is they are constructed: every one of them has a successor, 0 is the only one them that is not a successor, it starts the whole construction off from nothing. There is lot to think about here, like why not just write them down and put them in a book and be done with it? At least they have to have names don’t they, so write them down. My thoughts, I don’t have a word for them.

Seems to me there is a big education question here. Words before thought require authority. Is that the basis of school education now?

Chester Draws

Dick Fuller, the Sapir Whorf hypothesis is what I think you are wanting. I don’t believe that it is true, incidentally — we sure can think about things for which we have no word.

But I’ll put good money that you are extremely careful about your terminology when you teach. I would hope, for example, that you don’t talk about a parabola as a “line”, although students often do, and that you don’t mix the words “expression” and “equation”, although students often do.

Teachers are careful because using the correct words aids understanding, because it limits confusion. Sure you can think about things without the “correct” words, but it’s not a good idea. If a student uses “equation” for an expression, then they are going to struggle when the teacher says “you need an equation before you can solve”.

In my teaching I have two different sets of words, and I make it explicit which group words fall into. The key ones that I expect students to know and use — equation, line, curve, tangent, etc — and the ones that they should recognise when I or the book use them — vertex, coefficient, equivalent etc.

⚠️ Warning: Dangerously high levels of self-regard in this area. ⚠️

I know it’s terribly old-fashioned of me to expect students to learn stuff.

But no-one suggested that the way to do it is word lists, especially not at the start of a topic. I don’t teach terms before or after I introduce the concepts. I teach it before, after and during, so they get exposure over time.

This whole modern “we need to teach them to behave like Mathematicians” comes with a cost — Mathematicians are some of the most precise people about terminology in the world! If we allow them to be woolly about key terminology, then we allow them to be woolly about Maths.

Dan Meyer

Alex

Dick Fuller

I like the way Dan gets to the heart of a matter. In this context I am a weird learner trying to figure out why students have problems with negative numbers and fractions. The problem has been around forever, it must be too deeply buried in instruction to find. My approach has been to try and assess the gap between what is now taught and what could demystify the three pre-limit number systems, nats, ints, and rats. Constructing them seems like a good bet. This is not the place for the full story. If anybody is interested, I’ll start putting it up on my blog. What instruction does now is left to you. Construction is straight forward:
nat :every nat except 0 is the successor of another nat. Nats are ordered and unsigned (contradict/discuss). Can a nat have an “opposite”? What is subtraction?

Int: the (additive) difference from the nat 0 to the nat 1 is the int 1, from nat 1 to nat 0 is -1.
0 to 1 to 0 = 0 to 0; 1 to 0 to 1 = 1 to 1. Is 0 to 0 = 1 to 1? What is subtraction of -6? You are grounded, is touching power line with -6000 volts better than touching one with 6000 volts?

Construction is not difficult. It displays concepts and rewards reasoning. It is not the usual vocational approach now taken to avoid the need to think. It reveals itself, it does not require acquiescence to authority, it is not in the standards.

For the sake of argument let’s say: school mathematics is now taught by teachers required to accept the authority of standards, students required to accept material whose validity derives from authority, not reasoning. If that were true, would we expect success with students who wanted to think for themselves? And how about those with a deep distrust of authority?

Bringing this back to earth: I tried to talk about constructing numbers with my grandson. He wasn’t interested, said his teacher did math another way, she put negative numbers on other side of 0 from the positive whole ones.

Karise

We have been discussing set theory in my college introductory math class. Today, I asked them to make a Venn diagram that would show the relationship between different types of numbers. They worked for a little while in their groups and then we worked together as a class until we had a Venn diagram that we all liked and understood. One of my students asked about the relevance of this particular exercise. So, I said to her “I’m thinking of a number between 1 and 100. I’ll let you ask ten yes or no questions to figure out my number. If you guess a number, I’ll tell you if my number is higher or lower.”

Nice!

She immediately asked if my number was a whole number. When I answered “No.” I saw light bulbs come on for at least half the class. It was awesome!

She didn’t end up actually guessing my number, but there was a lot of great conversation that came out of her guessing process. One of the questions she asked after her first was “Is it a natural number?” Several of her classmates immediately spoke up and expressed their disappointment that she had wasted a question. In the end, most of the students in my class came to understand the distinction between different types of numbers and why that might be important. Thanks for the great idea!

Pat Hennelly

Love this idea to teaching students about how extensive the real numbers are. I’m thinking about this in the context of the class, and I am most curious about how this could be done to get an entire class thinking about the real numbers as a whole. This is something that can be done easily one-on-one with a student, but I am trying to figure out how to get the entire class involved in this wager. The video you showed definitely could help with that, but I feel as if it would be hard to have the whole classroom engaged in this. Any suggestions on how to accomplish the entire class being engaged?

Dan Meyer

Interesting thought, Pat. With Desmos, it’d be possible to create an activity where the teacher sets the secret number and then every student who has the code to the activity makes their own guesses. I’ll have to give that a try.

Eve P

I love this game and the idea of using it to introduce rational numbers to students. I am thinking even more basic than that is helping students with number sense in general. I think that this fun activity can get them to think about what numbers represent and how they are their own language with various terms to represent different types of numbers. I hope this can be a way for students to better understand numbers in general and get them curious about the ways in which they work. Thank you!